The form f = (a, b, c) of positive discriminant d is called reduced if the roots of azed-bw-Fc=o are of opposite signs, if one root
is numerically
I. Although every form is equivalent to a reduced form and the number of reduced forms of dis criminant d is finite, usually each reduced form is equivalent to several other reduced forms, and the number of the latter de pends upon the periodicity of the continued fraction for col. There is also another reason why the theory is more complicated than that for negative discriminants (see the topic positive binary quadratic forms). When d>o, the number of linear trans formations with integral coefficients which leave f unaltered (called automorphs of f) is infinite, in fact one for each set of integral solutions 1, u of the Pellian equation where a. is the divisor of f. The difficult point is to prove the existence of solutions T, U in least positive integers such that if t, u be any further solution in positive integers then t> T, u> U. It is then easily proved that all sets t, u of integral solutions are given by where Ea is the sum of the quadratic residues < z (I) -1) of p and E33 is that of the non-residues, while T, U give the solution of
-pw= I in least positive integers, and k takes those values from I, 3, 5, , 4p- I of which p is a non-residue.
N be the number of proper representations of m by
Kronecker proved in 186o by use of series for elliptic functions that N = 24 F(m) - I 2 G(m), where G(m) is the number of classes of binary forms of determinant - m, while F(m) is that of the forms in which at least one of a and c is odd. This is equiva
lent to a result due to Gauss (1800. Others found that , where (s/m) is Jacobi's symbol and is zero if s and m have a common factor. (See FORMS, ALGEBRAIC.) Positive Quadratic Forms in k Variables.Let N(n, k) denote the number of representations of n as a sum of k squares. Then N(4r+ r, 6) = I 2S,
20S, with s
where
ranges over the divisors of type 4k +I, and
over the divisors 4k+3. For m odd, N(m, 8) =
where d ranges over all divisors of m, and N(2rm, 8)
While N(4r+3, io) =
the expressions for N(n, 1o) are complicated if nX4r+3. For m odd and r >o, The numbers N(n, 5) and N(n, 7) have been expressed as sums involving Jacobi's symbol (s/ n), and in various other ways.
In a series of 18 papers in his journal for 1858-65, Liouville stated without proof many formulae and applications to the present topic. For example, he employed any function f(x, y) Cahen, Theorie des Nombres (vol. i., 1914, vol. ii., 1924) ; G. B. Mathews, Theory of Numbers (1892, out of print) ; L. E. Dickson, Theory of Numbers.
Texts in German on algebraic numbers: H. Weber (1891), J. Konig (1903), P. Bachmann (1905), H. Minkowski (1907), J. Sommer (1907), K. Hensel (5908), E. Landau (1918), E. Hecke (1923).
(L. E. D.)